MA 15200 - Purdue University
MA 15200 Lesson 28 Section 4.2
Remember the following information about inverse functions.
1. In order for a function to have an inverse, it must be one-to-one and pass a horizontal line test.
2. The inverse function can be found by interchanging x and y in the function’s equation and solving for y.
3. If [pic]. The domain of f is the range of [pic] and the range of f is the domain of [pic].
4. The compositions [pic] both equal x.
5. The graph of [pic] is the reflection of the graph of f about the line [pic].
Because an exponential function is 1-1 and passes the horizontal line test, it has an inverse. This inverse is called a logarithmic function.
I Logarithmic Functions
According to point 2 above, we interchange the x and y and solve for y to find the equation of an inverse function.
[pic] exponential function
[pic] How do we solve for y? There is no way to do this.
Therefore a new notation needs to be used to represent an inverse of an exponential function, the logarithmic function.
Definition of Logarithmic Function
[pic]
The function [pic] is the logarithmic function with base b.
The equation [pic] is called the logarithmic form and the equation [pic] is called the exponential form. The value of y in either form is called a logarithm. Note: The logarithm is an exponent.
Exponential Form Logarithmic Form
[pic] [pic]
argument base exponent exponent base argument
Ex 1: Convert each exponential form to logarithmic form and each logarithmic form to exponential form.
[pic]
[pic]
II Finding logarithms
Remember: A logarithm is an exponent.
Ex 2: Find each logarithm.
[pic]
[pic]
[pic]
III Basic Logarithmic Properties
1. [pic] Since the first power of any base equals that base, this is reasonable.
2. [pic] Since any base to the zero power is 1, this is reasonable.
The exponential function [pic] and the logarithmic function [pic] are inverses.
[pic]
This leads to 2 more basic logarithmic properties.
3. [pic] This is a composition function where [pic]. [pic] (the exponent)
4. [pic] This is a composition function where [pic]. [pic](the number or argument)
Ex 3: Simplify using the basic properties of logarithms.
[pic]
Ex 4: Simplify, if possible.
[pic]
IV Graphs of Logarithmic Functions
Below is a graph of [pic].
[pic]
If you imagine the line y = x, you can see the symmetry about that line.
Below are both graphs on the same coordinate system along with y = x.
[pic]
Characteristics of a logarithmic Graph:
The inverse of this function, [pic], has a graph with the following characteristics.
1. The x-intercept is (1, 0).
2. The graph still is increasing if b > 1, decreasing if 0 < b < 1.
3. The domain is all positive numbers, so the graph is to the right of the y-axis. (The range is all real numbers.)
4. The y-axis is an asymptote.
V Common Logarithms
A logarithmic function with base 10 is called the common logarithmic function. Such a function is usually written without the 10 as the base.
[pic]
A calculator with a key can approximate common logarithms.
Put the number (argument) in the calculator, press the common log key.
Ex 5: Find each common logarithm without a calculator.
[pic]
Ex 6: Use a calculator to approximate each common logarithm. Round to 4 decimal places.
[pic]
Using the basic properties with base 10, we get the following properties.
1. [pic]
2. [pic]
3. [pic]
4. [pic]
VI Natural Logarithms
A logarithm function with base e is called the natural logarithmic function. Such a function is usually written using ln rather than log and no base shown.
[pic]
A calculator with a key can approximate natural logarithms.
Put the number (argument) in the calculator, press the natural log key.
Ex 7: Use a calculator to approximate each natural logarithm. Round to 4 decimal places.
[pic]
Using the basic properties with base e, we get the following properties.
1. [pic]
2. [pic]
3. [pic]
4. [pic]
VII Modeling with logarithmic functions
The function [pic] gives the percentage of adult height attained by a boy who is x years old.
Ex 8: Approximately what percentage of his adult height has a boy of age 11 acheived?
(Notice: This model uses a common log.) Round to the nearest tenth of a percent.
The function [pic] models the temperature increase in degrees Fahrenheit after x minutes in an enclosed vehicle when the outside temperature is from 72° to 96°.
Ex 9: Use the function above to approximate the temperature increase after 45 minutes. Round to the nearest tenth of a degree.
-----------------------
[pic]
Remember that bases must be positive and the argument values (the numbers) must be positive.
(2, 1)
(1, 2)
log
ln
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